Department of Electronics Nanoelectronics 02 Atsufumi Hirohata 12:00 17/January/2014 Friday (D/L 002)

Quick Review over the Last Lecture Nano-scale miniaturisation : 4 reduction of ( effective electron paths ) 4 reduction of ( electron scattering ) ( faster ) operation 8 nano-fabrication ; ( complicated ) processes ( higher ) cost ( larger ) distributions in device properties 8 ( leakage ) current 8 ( Joule ) heating 8 electron ( confinement ) Electron transport : ( diffusive ) transport ( electron scattering ) ( ballistic ) transport ( negligible electron scattering )

Contents of Nanoelectronics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scalar and vector potentials III. Basics of quantum mechanics (04 ~ 06) IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) V. Nanodevices (08, 09, 12, 15 ~ 18) Lecture notes and files can be found at

02 Maxwell Equations Electromagnetic field Origins of an electromagnetic field Boundary conditions of an electromagnetic field

Maxwell Equations Maxwell equations are proposed in 1864 : E : electric field, B : magnetic flux density, H : magnetic field, D : electric flux density, J : current density and : charge density * Supplemental equations for materials : Definition of an electric flux density Definition of a magnetic flux density Ohms law

Maxwell Equations - Origins of an electromagnetic field Maxwell equations : For a time-independent case, Ampères law H i i dHdH Biot-Savart law Gauss law : An electrical charge induces an electric field. E

Faradays law of induction : Maxwell Equations - Boundary conditions of an electromagnetic field Maxwell equations : NS current force magnetic field current force magnetic field Gauss law for magnetism : Conservation of magnetic flux * N S

Maxwell Equations in Free Space Maxwell equations : In free space (no electron charge, and, and : constant), By differentiating the first equation with t and substituting the second equation,

Maxwell Equations in Free Space (Cont'd) Here, the left term can be rewritten as Similarly, For an ideal insulating matrix, * Electric field Magnetic field Propagation direction Electromagnetic wave propagation speed : in a vacuum,

Electromagnetic Wave *

Essence of the Maxwell Equations Maxwell equations unified electronics and magnetism : Further unification with the other forces ElectronicsMagnetism Electron chargeSourceMagnetic dipole moment Force (Coulombs law) Field Potential Flux (Gauss law) Einsteins theory of relativity

Michelson-Moley Experiment In 1881, Albert A. Michelson and Edward W. Morley precisely designed experiment to prove the presence of Ether : No interference between parallel / perpendicular to Ether flow * Ether was believed exist as a matrix to transfer an electromagnetic wave. No sign of Ether No relative speed !

Einstein's Theory of Relativity In 1905, Albert Einstein proposed the theory of special relativity : Speed of light (electromagnetic wave) is constant. * Lorentz invariance for Maxwells equations (1900) Poincaré proved the Lorentz invariance for dynamics. Lorentz invariance in any inertial coordinates

Unified Theory beyond the Maxwell Equations Big bang and Grand Unification Theory * Big bang s s s 0 Gravity Strong nuclear force nucleus Weak nuclear force -decay Electromagnetic force Weinberg-Salam Theory Maxwell Equation